IEEE Robotics & Automation Magazine - September 2019 - 55

Alternatively, the toolbox can be inspected and run completely
contained on Code Ocean [20].
Generating Joint Models
The Compliant Joint Toolbox comprises linear models of
both the mechanical actuator subsystem and the electrical
actuator subsystem as well as a number of parasitic and
nonlinear effects. This section details how to provide the
parameters for such dynamic effects and how to generate
model classes from them. The following sections provide a
number of code examples, which can be found in
Take_a_Tour.m.
Generic Model Implementation
The linear electrical and linear mechanical subsystem models
(Table 1) of a compliant electrical actuator form the core of
the Compliant Joint Toolbox. Nonlinear terms use the states
of these subsystems and modulate their input-output behavior, which allows the capture of a broad range of practically
relevant nonlinear dynamic effects.
The Electrical Subsystem
The most common electrical drive in torque-controlled
robotic actuators is the brushless dc motor, which can be
operated such that the actual three-phase motor dynamics are
well described by a single-phase approximation. The governing parameters are the electrical resistance and inductance. In
torque-controlled electrical actuators, the inductance is typically designed to be low. As a consequence, the electrical time
constant becomes very small (. 10 -6 s) compared to the
mechanical time constant (. 10 -3 s) . Unless it is specifically
intended to analyze the current control performance or its
implications for higher-level controllers, the electrical dynamics can be neglected with respect to the mechanical time constant. This substantially shortens the simulation time. Hence,
a static model is used by default in building actuator models.
The "Model Generation" section describes how to switch to a
dynamic model.
The Mechanical Subsystem
The mechanical subsystem is modeled as a chain of rotating
masses interconnected via massless spring-damper elements,
as depicted in Figure 3. The electrical drive rotor has an inertia I m, which experiences a damping d m with respect to
ground. The gearbox contributes an inertia I g and can be
compliant with a linear stiffness k g and internal damping
d mg . Gear friction with respect to ground is captured by d g .
The second elastic element is represented by a massless torsional spring with linear stiffness k b and internal material
damping d gl . Finally, the rotary inertia I l models the load
with frictional damping d l . The motor, gearbox, and load
angles are denoted by q m, q g , and q l, respectively. The
torques acting on the motor, gearbox, and load are x m, x g ,
and x l , respectively.
Deriving the linear equations of motion for this three-mass
system from first principles is straightforward and can even be

found in many textbooks
on control or structural
The toolbox was an
dynamics, such as [1]. The
Compliant Joint Toolbox
essential tool in the
features several variants of
this general model strucpreparation of publications
ture, such as a rigid gearbox, complete rigidity (a
on modeling, observer
single moving mass with
friction), and fixed-output
design, torque, and
configurations. In the latter, the load motion is
impedance control.
defined by an external
source, effectively allowing
the connection of the actuator model to the complex articulated robot dynamics. Load
motion can be zero to emulate a locked actuator output or,
equivalently, infinitely high load inertia. This last scenario is
often used for torque controller design and analysis [2]-[4].
The Compliant Joint Toolbox implements the linear
mechanical dynamics in state-space form. The joint
model has, in total, two inputs and generally seven
outputs. The two inputs are the motor current and a

Figure 2. The browser example (cjtExamples.m) allows
browsing through available MATLAB and Simulink offerings to
inspect or directly run them.

qm,

qg,

ql,

kb
Ig

dmg
dg

dgl

Figure 3. The linear mechanical system at the core of the
Compliant Joint Toolbox.

SEPTEMBER 2019

IEEE ROBOTICS & AUTOMATION MAGAZINE

IEEE Robotics & Automation Magazine - September 2019

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